Supplementing Peter's answer to Streicher's K-finiteness question, I recall Prop. 7.4 on p. 97 of SLNM #753, which states, for presheaf topoi E = (C^op, Sets), that, with E_Kf the full subcategory of K-finite E-objects: E_Kf is balanced iff it's a topos iff each K-finite is decidable iff C is a "2-way" category iff ... . Streicher's >--> sure isn't 2-way, hence ... . The rest of that 20 year old report on my student Acun~a's thesis with me is also fun. -- Fred

Sorry, I used K/S for an abbreviation of what was called Kuratowski until someone pointed out that it was due to Sierpinski :an object whose mark belongs to the smallest sub-semilattice of its power set which contains the singleton map, or in case there is an NNO an object which in a suitable sense is locally enumerable by the segment under a section of the NNO . While the K/S definition is right for the construction of the object classifier over an arbitrary base topos (as Gavin showed) and hence for classifiers for various kinds of finitary algebras over an arbitrary base topos, still the theory of it in the last 25 years of topos theory seems to mainly be justified by formal analogy and/or independence relative to abstract set theory (=topos with choice). However there are important uses of "finiteness" in algebraic geometry and differential topology (where topos theory after all started) : Consider a ringed topos E,R . For example, the sheaves on an algebraic variety or on a Cinfty manifold. Within the abelian category of R-modules in E, we need to single out two important subcategories FAC (Serre 1955)=coherent sheaves..these tend to be an abelian subcategory and tend to vary covariantly as one E,R is mapped to another E',R' (thus give rise to an extensive K-homology) and vector bundles , which one thinks of as a finite-dimensional vector space varying smoothly over the base space of E ,so they cry out for internalization ; in algebraic geometry these are identified with locally FINITELY free R-modules... they vary contravariantly with E,R (so give rise to K-cohomolgy rigs which act on the FACs,ie intensives acting as densities on the extensives; with further conditions on E,R one can at the level of the riNgs generated by these rigs define a sort of Radon/Nikodym derivative via an alterating sum of Tors , but in general the covariant abelian category FAC and the contravariant tensored category Vect are distinct...The "derived category" of E,R (now allegedly replacing homological algebra in complex analysis and C*-algebra theory) should be the derived category of one of these two linear categories (here I mean dc in the linear sense..nonlinear "derived categories" are more like the stable homotopy of E)) Already the intuitionists speculated about (in effect) subobjects of K/S objects, and it seems we need something of the sort perhaps a category of finites closed under subquotient in order to define the notion of eg finitely-generated R-module in a way which not merely mimics abstract set theory but actually captures the vector bundles . Perhaps it will be easier if E itself satisfies a noetherian condition. It would be best if the desired content could be entirely int- ernalized to E,R but perhaps it is really relative to a base S,K..but perhaps without restriction on S ?? I hope this clarifies the problem. Sincerely Bill

Concerning Peter Johnstone's clarification: Of course I didn't mean that the object classifier could be constructed without an internal parameterizer for the finite objects in the base S .... but what exactly are the finite objects ? While the classifier as a topos is determined by the 2-category of bounded S-toposes , the site for it isn't. I was under the impression that an internal category parameterizing the objects which are both K-finite and separable(=decidable) could be used (while internal presheaves on "all" K=finites would presumably be much bigger..what does IT classify ?) Anyway my point was that at any rate no further extension of the notion of finiteness is needed for classifying in that sense the objects or the group objects in S-toposes, whereas by contrast it seems that to give the mathematically correct notion of "vector space for which there exists a finite basis" does need such an extension. In any topos, a subobject of a nonnon sheaf is always separable ; when is the converse true ? Perhaps there is an internal topos object V which is largest with respect to being fully embedded in the given topos E while at the same time having A as its subtopos of internal nonnon sheaves. Here by A is meant the Boolean internal topos mentioned above which parameterizes the separable K-finites of E (Fred recalled Acunya's work showing among other things that it is Boolean) and to say that V "is" fully embedded in E has sense for any internal category with a terminal object , namely we require that the canonical parametrized (="indexed") functor from V to E is an equivalence E(X,V)--> E/X for each X. The latter functor is defined by merely pulling back the fibration 1/V--> V of pointed objects in V. When the answer to the above question is affirmative, Johnstone's locally separable reflection Vsubqd will consist of subquotients and the K-finites may fit in . It seems that the inclusion of A in V will preserve sums but only certain epis. The idea is that V can't be too large since the inverse to the inclusion will enrich it in A. On Wed, 15 Jan 1997, categories wrote:

Date: Wed, 15 Jan 97 10:19 GMT From: Dr. P.T. Johnstone <P.T.Johnstone@pmms.cam.ac.uk>

Not an answer to Bill's question (which I agree is an important one), but a minor correction. Bill wrote:

While the K/S definition is right for the construction of the object classifier over an arbitrary base topos (as Gavin showed) and hence for classifiers for various kinds of finitary algebras over an arbitrary base topos,

It isn't, and he didn't. Gavin used finite cardinals to construct the object classifier over an arbitrary base topos with NNO (and I subsequently extended the construction to finitary algebraic theories), but it doesn't work over a topos without NNO (and in particular it can't be made to work using K-finiteness). Andreas Blass showed that the existence of an object classifier for toposes over E implies that E has a NNO.

Incidentally, I think it is correct to give credit to Kuratowski for the notion of K-finiteness. It's true that Sierpinski's paper was earlier, but his definition was a "global" one (i.e. he defined the class of all finite sets as the sub-semilattice of the universe generated by he singletons), whereas Kuratowski made the crucial observation that the finiteness of a particular set X can be determined locally (i.e. within the power-set of X), without which the notion could never have been imported into topos theory.

Peter Johnstone